- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 08:58:49
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=EMSS&vol=4&iss=1&update_since=2024-03-29
EMS Surveys in Mathematical Sciences
EMS Surv. Math. Sci.
EMSS
2308-2151
2308-216X
General
10.4171/EMSS
http://www.ems-ph.org/doi/10.4171/EMSS
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
4
2017
1
A survey on the singularities of spherical varieties
Boris
Pasquier
Université de Montpellier, MONTPELLIER CEDEX 5, FRANCE
Spherical varieties, singularities, minimal model program
We list combinatorial criteria of some singularities, which appear in the Minimal Model Program or in the study of (singular) Fano varieties, for spherical varieties. Most of the results of this paper are already known or are quite easy corollary of known results. We collect these results, we precise some proofs and add few results to get a coherent and complete survey.
Algebraic geometry
1
19
10.4171/EMSS/4-1-1
http://www.ems-ph.org/doi/10.4171/EMSS/4-1-1
Invitation to $H$-systems in higher dimensions: known results, new facts, and related open problems
Armin
Schikorra
Albert-Ludwigs-Universität Freiburg, FREIBURG, GERMANY
Paweł
Strzelecki
University of Warsaw, WARSZAWA, POLAND
$n$-Laplacian, regularity of solutions, $H$-system
In this paper, we discuss two well-known open problems in the regularity theory for nonlinear, conformally invariant elliptic systems in dimensions $n ≥ 3$, with a critical nonlinearity: $H$-systems (equations of hypersurfaces of prescribed mean curvature) and n-harmonic maps into compact Riemannian manifolds. For $n = 2$ several solutions of these problems are known but they all break down in higher dimensions (unless one considers special cases, e.g. hypersurfaces of constant mean curvature or manifolds with symmetries). We discuss some of the known proofs and hint at the main difficulties. We also state a few new results (such as positive answers for all solutions of class $W^{n/2,2}$ for even $n$, instead of $W^{1,n}$ and list some open questions of independent interest — including specific endpoint variants of the Coifman–Rochberg–Weiss theorem, addressing the boundedness of commutators of fractional and singular integrals with multiplication by bounded functions of class $W^{1,n}$ — that would lead to solutions of these two problems.
Partial differential equations
Global analysis, analysis on manifolds
21
42
10.4171/EMSS/4-1-2
http://www.ems-ph.org/doi/10.4171/EMSS/4-1-2
High-dimensional statistics, with applications to genome-wide association studies
Peter
Bühlmann
ETH Zentrum, ZÜRICH, SWITZERLAND
De-sparsified Lasso, hierarchical multiple testing, Lasso, $\ell_1$-norm regularization, sparsity
We present a selective review on high-dimensional statistics where the dimensionality of the unknown parameter in a model can be much larger than the sample size in a dataset (e.g. the number of people in a study). Particular attention is given to recent developments for quantifying uncertainty in high-dimensional scenarios. Assessing statistical uncertainties enables to describe some degree of replicability of scientific findings, an ingredient of key importance for many applications. We also show here how modern high-dimensional statistics offers new perspectives in an important area in genetics: novel ways of analyzing genome-wide association studies, towards inferring more causal-oriented conclusions.
Statistics
Probability theory and stochastic processes
45
75
10.4171/EMSS/4-1-3
http://www.ems-ph.org/doi/10.4171/EMSS/4-1-3
Rigidity of spreadings and fields of definition
Chris
Peters
Technical University Eindhoven, EINDHOVEN, NETHERLANDS
Belyı curves, Beauville surfaces, fields of definition, Higgs fields, rigidity, Shimura varieties, spreads
Varieties without deformations are defined over a number field. Several old and new examples of this phenomenon are discussed such as Bely˘ı curves and Shimura varieties. Rigidity is related to maximal Higgs fields which come from variations of Hodge structure. Basic properties for these due to P. Griffiths, W. Schmid, C. Simpson and, on the arithmetic side, to Y. André and I. Satake all play a role. This note tries to give a largely self-contained exposition of these manifold ideas and techniques, presenting, where possible, short new proofs for key results.
Algebraic geometry
Number theory
77
100
10.4171/EMSS/4-1-4
http://www.ems-ph.org/doi/10.4171/EMSS/4-1-4